Cotangent Lines

by Kyle Schultz

The Problem: Find two linear functions f(x) and g(x) such that their product h(x) = f(x).g(x) is tangent to each of f(x) and g(x) at two distinct points.

Click here to see an example.

I will define f and g as cotangent lines:

Definition: Two distinct linear functions, f and g, are cotangent if their graphs are tangent to the parabola that is the graph of f.g.

To determine the nature of cotangent lines, I used an algebraic approach. First I defined f, g, and f.g as follows:

f(x) = ax + b

g(x) = cx + d

fg(x) = (ax + b)(cx + d)

Next , I set f = fg to find the values of x for which the first line intersects the parabola.

ax + b = (ax + b)(cx+d)

0 = (ax + b)(cx+d) - (ax +b)

0 = (ax + b)(cx + d - 1)

ax + b = 0 or cx + d - 1 = 0

This leads to the solutions:


If the line is tangent to the parabola, there is only one point of intersection. By setting the two values of x equal to each other, an equation is formed relating the values of a, b, c, and d.

-bc = a - da

a = da - bc

By a similar process, setting cx + d = (ax + b)(cx + d) will yield the equation:

c = bc - da

This yields a very important fact: a = -c. We can conclude the following:

Theorem: The slopes of two cotangent lines are opposites.

Substituting -a for c in either equation yields another important fact:

a = da - b(-a)

a = da + ba

a = a(d + b)

1 = d + b

Thm: The sum of the y-intercepts of two cotangent lines is 1.

Now, given any linear function, it is quite simple to find its the function for its cotangent line. Try this example:

What is the function for the line cotangent to f(x) = 5x - 4?

Click here to view the solution.


Calculus: Use the first derivative of fg to verify the two theorems stated above.

Click here for a hint.

Cotangent Parabolas: Is there a second parabola that is tangent to two cotangent lines?

Click here for an introduction.

EMAT 6680 Class Page for Kyle Schultz